An Efficient Rectilinear Steiner Minimum Tree Algorithm Based on Ant Colony Optimization* Yu Hu, Tong Jing, Xianlong Hong, Zhe Feng Xiaodong Hu, Guiying Yan Tsinghua University Institute of Applied Mathematics, CAS Beijing 100084, P. R. China Beijing 100080, P. R. China Email: matrix98@mails.tsinghua.edu.cn Email: xdhu@public.bta.net.cn Abstract--The rectilinear Steiner minimum tree (RSMT) problem is one of the fundamental problems in physical design, especially in routing, which is known to be NP-complete. This paper presents a practical heuristic for RSMT construction based on ant colony optimization (ACO). This algorithm has been implemented on a Sun workstation with Unix operating system and the results have been compared with the GeoSteiner 3.1 and a recent work using batched greedy triple construction (BGTC). Experimental results show that our algorithm, named ACO-Steiner, can get a very short run time and keep the high performance. Keywords: rectilinear Steiner minimum tree (RSMT), routing, physical design, ant colony optimization (ACO) I. INTRODUCTION Routing plays an important role in very large scale integrated circuit/ultra large scale integrated circuit (VLSI/ULSI) physical design [1]. The rectilinear Steiner minimum tree (RSMT) problem is one of the fundamental problems in routing [2]. However, Garey and Johnson [3] prove that the RSMT problem is NP-complete, indicating that a polynomial-time algorithm to compute an optimal RSMT is unlikely to exist. So, many helpful algorithms continue to focus on the RSMT problem to get high efficiency. Ref. [4] gave an extensive survey of RSMT heuristics in 1992. Kahng and Robins [5] introduced the Batched Iterated 1-Steiner (Bl1S) heuristic with an average improvement over the minimum spanning tree (MST) on terminals of almost 11%. Two good works appeared recently. Kahng and Mandoiu et al [6] proposed a batched greedy triple construction (BGTC) algorithm, which speedups the run time of Zelikovsky’s algorithm [7] while keeping its performance. Zhou [8] introduced the spanning graph as a base for MST and then constructed the RSMT from the MST. Another work is the O(nlogn) algorithms proposed in [9], which is in octilinear plane instead of rectilinear. Warme et al released the GeoSteiner [10, 17], which is an exact algorithm. The shortcoming of GeoSteiner is the long run time. So, There is room for * This work was partially supported by NSFC under Grant No.60373012, Hi-Tech Research and Development (863) Program of China under Grant 2002AA1Z1460, SRFDP of China under Grant No.20020003008, and NSFC under Grant 60121120706. obtaining high efficiency. The main contribution of this paper is to propose a practical heuristic, called ACO-Steiner, to construct a RSMT, by which we can get a very short run time and keep the high performance. When the number of terminals is no more than 50, ACO-Steiner can achieve exact results (better than BGTC) but keeping the fast speed. When the number of terminals is more than 50, ACO-Steiner can achieve near optimal results (within 1% wire length increments compared with GeoSteiner) but keeping the very short run time. The rest of this paper is organized as follows. In Section II, we introduce the ant colony optimization (ACO) and some basic definitions of RSMT problem. In Section III, the ACO-Steiner heuristic is described in detail. Section IV gives performance improvements based on some special strategies. Then, Section V shows the experiment results. Finally, Section VI concludes the whole paper. II. PRELIMINARIES A. ACO algorithm As we know, ants live in colonies and have evolved to exhibit very complex patterns of social interaction. Besides the simplistic behavior of individual ants, they can communicate with one another through secretions called pheromones, and this cooperative activity of the ants in a nest gives rise to an emergent phenomenon known as swarm intelligence. ACO algorithms are a class of algorithms that mimic the cooperative behavior of real ant behavior to achieve complex computations [11]. The ACO consists of multiple iterations. In the iteration of the algorithm, one or more ants are allowed to execute a move, leaving behind a pheromone trail for others to follow. An ant traces out a single path, probabilistically selecting only one edge at a time (in a graph), until an entire solution is obtained. Each separate path can be assigned a cost metric, and the sum of all the individual costs defines the function to be minimized by ACO [12]. The main flow of ACO algorithm is shown in Fig.1. B. Basic definitions of RSMT problem The RSMT problem is described as follows [13]. Given a set T of n points called terminals in the plane, find a set S of additional points called Steiner points such that the length of a rectilinear minimum spanning tree of